Abstract
The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.
Highlights
In real world, many fundamental laws of physics and chemistry can be formulated as differential equations
Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients
As a powerful Mathematical tool for modeling many of natural models in applied sciences, one can refer to partial differential equations (PDEs)
Summary
Many fundamental laws of physics and chemistry can be formulated as differential equations. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. Diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As a powerful Mathematical tool for modeling many of natural models in applied sciences, one can refer to partial differential equations (PDEs). PDEs are used to formulate problems involving functions of several variables, and are usually difficult to solve. It is necessary applying high accurate numerical methods
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